# Nonlocal statistical field theory of dipolar particles in electrolyte solutions

## TL;DR

This paper develops a nonlocal statistical field theory for dilute electrolyte solutions containing dipolar particles, deriving generalized equations and analytical expressions for electrostatic potential and solvation free energy, including in salt-free and salt solutions.

## Contribution

It introduces a novel nonlocal field theoretical framework that generalizes the Poisson-Boltzmann-Langevin equation for dipolar particles with arbitrary charge distribution functions.

## Key findings

- Derived a non-linear integro-differential equation for the electrostatic potential.
- Obtained analytical expressions for electrostatic potential and solvation free energy for specific PDFs.
- Provided a general formula for the electrostatic free energy within the RPA approximation.

## Abstract

We present a nonlocal statistical field theory of a dilute electrolyte solution with small additive of dipolar particles. We postulate that every dipolar particle is associated with an arbitrary probability distribution function (PDF) of distance between its charge centers. Using the standard Hubbard-Stratonovich transformation, we represent the configuration integral of the system in the functional integral form. We show that in the limit of a small permanent dipole moment, the functional in integrand exponent takes the well known form of the Poisson-Boltzmann-Langevin (PBL) functional. In the mean-field approximation we obtain a non-linear integro-differential equation with respect to the mean-field electrostatic potential, generalizing the PBL equation for the point-like dipoles obtained first by Abrashkin et al. We apply the obtained equation in its linearized form to derivation of the expressions for the mean-field electrostatic potential of the point-like test ion and its solvation free energy in salt-free solution, as well as in solution with salt ions. For the 'Yukawa'-type PDF we obtain analytic relations for both the electrostatic potential and the solvation free energy of the point-like test ion. We obtain a general expression for the bulk electrostatic free energy of the solution within the random phase approximation (RPA). For the salt-free solution of the dipolar particles for the Yukawa-type PDF we obtain an analytic relation for the electrostatic free energy.